2.882 Complexity. April 20, 2005

Transcription

1 2.882 Complexity April 20, 2005

2 Complexity in AD Complexity Measure of uncertainty in achieving the desired functional requirements of a system Difficulty Relativity Information Ignorance

3 Four types of complexity in AD Complexity Measure of uncertainty in achieving FR Does uncertainty change with time? Time-independent Complexity Time-dependent Complexity Real complexity Imaginary complexity Combinatorial complexity Periodic complexity

4 Complexity: A measure of uncertainty in achieving the desired set of FRs of a system Time-independent real complexity Measure of uncertainty when the probability of achieving the functional requirements is less than 1.0 (because the common range is not identical to the system range) Time-independent imaginary complexity Uncertainty that arises because of the designer s lack of knowledge and understanding of a specific design itself Time-dependent combinatorial complexity Time-dependent combinatorial complexity arises because in many situations, future events cannot be predicted a priori. This type of time-dependent complexity will be defined as time-dependent combinatorial complexity. Time-dependent periodic complexity Consider the problem of scheduling airline flights. it is periodic and thus uncertainties created during the prior period are irrelevant. This type of timedependent complexity will be defined as time-dependent periodic complexity.

5 Time-independent Real Complexity Time-independent real complexity caused by system range s being outside of the design range. Real complexity ~ Information content Take u i as a random variable u i = 1 (success) with P(FR i = success) 0 (failure) 1-P(FR i = success) Information content: I(u i = 1) - log 2 P(FR i =success) dr 7 Design Range 6 p.d.f. Common 5 f(fr) Range, 4 A C 3 System Range, 2 p.d.f. f(fr) I(p) [bit] 1 dr l dr u FR P(FR sr =success) i 0 1

6 Time-independent Imaginary complexity Imaginary complexity ~ Ignorance Ignorance causes complexity. Types of ignorance Functional requirement Knowledge required to synthesize(or identify) design parameters Ignorance about the interactions between FRs and DPs p (probability of selecting a right sequence) For uncoupled design, p= 1 For decoupled design, p= z/n! For coupled design, p = 0

7 Time-dependent complexity Time-dependency Complexity Uncertainty in achieving a set of FR Complexity is time-dependent if 1) uncertainty (probabilistic) is time-dependent Time-varying system range 2) behavior of FR is time-dependent FR = FR(t) Combinatorial / Periodic complexity Uncertainty increases indefinitely : combinatorial complexity Uncertainty in one period is irrelevant to the next period : periodic complexity

8 Origins of complexity and reduction

9 Time-independent Minimize Real complexity by Eliminating source of variation Desensitizing w.r.t. variation Compensating error Eliminate Imaginary complexity by Achieving uncoupled design Identifying design matrix

10 Time-dependent complexity Combinatorial complexity Periodic complexity

11 Time-varying system range Detect changes in system range Prevent system range deterioration by design Bring the system range back into design range by re-initialization

12 Prevent system range deterioration by design By eliminating coupling between turn and grasp, one can effectively delay system range deterioration. Milled Flat end of the shaft Slot Milled Flat end of the shaft A Metal shaft Injection molded nylon Knob A (a) (b) Section view AA N. P. Suh, Axiomatic Design: Advances and Applications, 2001

13 Bring the system range back into design range: Re- initialization Example: Design of Low Friction Surface Dominant friction mechanism: Plowing by wear debris System range (particle size) moves out of the desired design range Need to re-initialize Figure removed for copyright reasons. Figure removed for copyright reasons. N. P. Suh and H.-C. Sin, Genesis of Friction, Wear, 1981 S. T. Oktay and N. P. Suh, Wear debris formation and Agglomeration, Journal of Tribology, 1992

14 Design of Low Friction Surface Periodic undulation re-initializes the system range Figures removed for copyright reasons. S. T. Oktay and N. P. Suh, Wear debris formation and agglomeration, Journal of Tribology, 1992

15 Example: Scheduling of mfg system Periodicity should be introduced & maintained to prevent the system from developing chaotic behavior Subsystem X Subsystem Y

16 Problem description B C Subsys X PTi or CTY Number of MvPki MvPli Station (sec) machines (sec) (sec) IN A D a X b c d Y IN Interface Subsystem Y Fastest speed* 70 seconds when X determines the system speed seconds when Y determines the system speed Objective Maximum utilization rate for the machine Y Constraint Transport from C to D must be immediate * Speed is measured by throughput time: shorter time means faster speed

17 3rd part ready at X 4th part ready at X 5th part ready at X 6th part ready at X FP X 1st part ready at X 2 FP X 3 FP X 4 FP X 2nd part ready at X Time, t

18 5th demand from Y Y process starts FP Y,5 z 6 D 5 z 4 FP Y,4 1st demand from Y Y process strats FP Y,1 z 2 FP Y,2 z 3 FP Y,3 3rd part ready at X 4th part ready at X 5th part ready at X 6th part ready at X FP X 2 FP X 3 FP X 4 FP X 1st part 2nd part ready at X ready at X Time, t

19 CT Y : 60sec - 60sec - 60sec - 60sec - 55sec - Y FINISH Y START Y FINISH CT Y = 60 sec Transport 1 2 Y START (next period) Max[ CT Y Y] ] = 65 sec Machine d1 1 2 Min[ CT Y Y] ] = 55 sec Machine c 3 4 Machine d2 Machine b 5 6 Machine c 3 4 Machine d1 Machine a 7 8 Machine b 5 6 Machine c SP=70sec 9 10 Machine a 7 8 Machine b New panel SP=70sec 9 10 Machine a New panel Figure 10. Steady state operation with 70 seconds sending period From Lee, Taesik. "Complexity Theory in Axiomatic Design." MIT PhD Thesis, 2003.

20 CT Y : 60sec - 60sec - 55sec - 65sec - 65sec Panel # Y FINISH Max[CT ] = 6 5 sec Y Y START Y FINISH Machine d1 1 2 Max[CT Y ] = 65 sec Y START 3 4 Machine d Machine c 3 4 Machine d1 7 8 Machine b 5 6 Machine c Machine a 7 8 Machine b X X X X X X X X No Transport SP = 95 sec New panel SP = 75 sec Figure 11 (b) Panel # 4 Y FINISH Max[ CT ] = 6 5 sec Y START Y Machine d1 1 2 Max[ CT Y ] = 6 5 sec Y FINISH Y START Machine d Machine c 3 4 Machine d1 7 Machine a 7 8 Machine b 5 6 Machine c 8 SP = 90 sec 9 10 X X X X New panel 9 10 Machine a X X X X No Transport Figure 11 (c) From Lee, Taesik. "Complexity Theory in Axiomatic Design." MIT PhD Thesis, 2003.

21 Average throughput time = ( )/4 = From Lee, Taesik. "Complexity Theory in Axiomatic Design." MIT PhD Thesis, 2003.

22 Single perturbation from subsystem Y causes incomplete period in downstream The system regains periodicity after the perturbation is removed but with undesirable performance Throughput time is seconds in average Slower than the system capability

23 Re-initialization scheme in scheduling Define a renewal event that imposes period u(0) = { u 0 (t), u 1 (t),, u k-1 (t), u k (t), u k+1 (t),, u N (t)} = {0, 0,, 0, 0, 0,, 0} : u(t-d) = {1, 1,, 1, 0, 1,, 1} u(t) = {1, 1,, 1, 1, 1,, 1} u(t+e) = {0, 0,, 0, 0, 0,, 0} = u(0) Scheduling activity is confined within such a period with a goal of maintaining periodicity Conditional renewal event t ini = t request if t request 70 sec ( FP X ) t ini = 70sec if t request < 70 sec

24 1st demand from Y Y process strats 2nd demand from Y Y process strats FP Y,1 3rd demand from Y Y process strats FP Y,2 t 2 4th demand from Y FP Y,3 t 3 5th demand from Y Y process starts D 4 t 4 D 5 4th part ready at X Y process starts t 5 5th part ready at X t 1 Time, t z 1 z 2 2nd part ready at X 3rd part ready at X z 6 Each period is independent (memoryless)

25 Conclusion Breakdown of functional periodicity results in sub-optimal throughput rate Periodicity should be introduced & maintained to prevent the system from developing chaotic behavior

26 Example: Cell division A cell has a mechanism to coordinate cycles of two subsystems such that the overall periodicity is maintained Break-down of functional periodicity leads to anomaly of cell division and further chaotic behavior of the system Maintaining functional periodicity in the cell cycle is an important functional requirement for cell division

27 Overview of the Cell Cycle Figure removed for copyright reasons. * Figure taken from Molecular Biology of the Cell, Alberts, Garland Science

28 Chromosome cycle & Centrosome cycle Figure removed for copyright reasons. * Figure taken from Molecular Biology of the Cell, Alberts, Garland Science

29 Importance of the correct number of chromosomes and centrosomes Centrosomal abnormalities Chromosome missegregation Aneuploidy Figure removed for copyright reasons. Figure 2 in Nigg, E. A. "Centrosome abberation: cause or consequence of cancer progression?" Nature Reviews Cancer 2 (2002): Figure removed for copyright reasons. * Figure taken from

30 Functional periodicity B C Machine X Figure removed for copyright reasons. See Figure 1 in Nigg, E. A. "Centrosome abberation: cause or consequence of cancer progression?" Nature Reviews Cancer 2 (2002): A IN Interface D Machine Y

31 Mechanism Cdk2 initiates both cycle ensuring one level of synchronization Cdk2 (S-Cdk) inactivate Rb inactivate E2F promote accumulation inactivate G1- Cdk G1/Scyclin G1/S- Cdk Mitogen-dependent mechanism (Extracellular signal) mutually inhibits mutually inhibits Hct1 APC Firing origin of replication Centriole split gene transcription inactivate S-cyclin S-Cdk mutually inhibits CKI:p27 Chromosome duplication Replicating DNA Check for completion of DNA replication Duplicate centrosome Check for completion Centrosome duplication DNA replication checkpoint ensures completion of chromosome duplication Prophase Prometaphase Metaphase Centrosome cycle lacks a mechanism to check its completion Check for spindle attachment Anaphase Telophase Cytokinesis

32 Conclusion A cell has a mechanism to coordinate cycles of two subsystems such that the overall periodicity is maintained Maintaining functional periodicity in the cell cycle is an important functional requirement for cell division Can pose questions with new perspective

33 Functional periodicity u(t) = {u 1 (t), u 2 (t),, u N (t)} Time-dependent FR Periodic There exist T i s.t. u(t i ) = u(t j ) with regular transition pattern Semi-periodic There exist T i s.t. u(t i ) = u(t j ) without regular transition pattern Aperiodic None of the above u 5 u 5 u 5 FR State u4 u 4 u 4 FR State u3 u 3 u 3 u2 u 2 u 2 FR State u 1 u 1 u 1 u 0 u 0 u 0 T 1 T 2 t T 1 T 2 t t (a) Periodic (b) Semi-periodic (c) Aperiodic

34 Uncertainty and functional periodicity P s (t) = P(u(t) = u*(t)) For periodic & semi-periodic FR(t), P s returns to one at the beginning of a new period Predictability of FR (Periodicity) fi (Predictability) (Unpredictability) fi (Aperiodicity) ~ (Aperiodicity) fi ~ (Unpredictability) Uncertainty in current period is independent of a prior period only if the initial state is properly established